A molecular dynamics simulation of polymeric brushes immersed in a melt of mobile polymer chains is presented. The brush height and monomer density profiles are presented for chains of length N grafted at one end to a solid surface immersed in a melt of polymers of the same type. As the chain length P of the free chains increases there is a crossover from a wet to a dry brush in agreement with scaling and self-consistent field theories. Since the interaction between all monomers are identical, this crossover is driven purely by entropic interactions. The simulation results are compared to earlier simulations in which the solvent is treated as a continuum. The explicit introduction of solvent molecules increases the equilibration time of the system by a factor of 40-100 even for small P. The dynamics of the brush chains were also studied. The relaxation time for the radius of gyration and end-to-end distance was found to be independent of P for small P in agreement with the prediction based on a hydrodynamic model by Johner and Joanny [J. Chem. Phys. 98, 1647 (1993)]. For larger P, comparable to an entanglement length, the relaxation time increases as the viscosity of the solvent crosses over from P-1 to P-3.